This chapter has introduced many of the concepts associated with
digital filters, such as signal representations, filter
representations, difference equations, signal flow graphs, software
implementations, sine-wave analysis (real and complex), frequency
response, amplitude response, phase response, and other related
topics. We used a simple filter example to motivate the need for more
advanced methods to analyze digital filters of arbitrary complexity.
We found even in the simple example of Eq.
(1.1) that complex
variables are much more compact and convenient for representing
signals and analyzing filters than are trigonometric techniques. We
employ a complex sinusoid
having three
parameters: amplitude, phase, and frequency, and when we put a complex
sinusoid into any linear time-invariant digital filter, the filter
behaves as a simple complex gain
, where the magnitude
and
phase
are the amplitude response and phase response,
respectively, of the filter.